Theorem intimasn2

Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020)

	Ref	Expression
Assertion	intimasn2	$$⊢B\in V\to \bigcap A\left[\left\{B\right\}\right]=\bigcap _{x\in A}x\left[\left\{B\right\}\right]$$

Proof

Step	Hyp	Ref	Expression
1		intimasn	$$⊢B\in V\to \bigcap A\left[\left\{B\right\}\right]=\bigcap \left\{y|\exists x\in Ay=x\left[\left\{B\right\}\right]\right\}$$
2		intima0	$$⊢\bigcap _{x\in A}x\left[\left\{B\right\}\right]=\bigcap \left\{y|\exists x\in Ay=x\left[\left\{B\right\}\right]\right\}$$
3	1 2	eqtr4di	$$⊢B\in V\to \bigcap A\left[\left\{B\right\}\right]=\bigcap _{x\in A}x\left[\left\{B\right\}\right]$$